Problem: $f(x) = \begin{cases} -4 & \text{if } x = 0 \\ -2x^{2}+2 & \text{otherwise} \end{cases}$ What is the range of $f(x)$ ?
First consider the behavior for $x \ne 0$ Consider the range of $-2x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $-2$ flips the range to $\{\, y \mid y \le 0 \,\}$ To get $-2x^{2}+2$ , we add $2$ If $x = 0$ , then $f(x) = -4$ , which eliminates $f(x) = 2$ from the range. The new range is $\{\, y \mid y < 2 \,\}$.